论文标题
hahn- banach扩展及其一些变体的独特性
Uniqueness of Hahn--Banach extensions and some of its variants
论文作者
论文摘要
在这项研究中，我们分析了Hahn-Banach扩展的独特性的各种加强和削弱。此外，我们考虑$ y $是$ x $的理想的情况。在这种情况下，我们研究了属性 - $（u）/（su）/（hb）$和属性 - $（k-u）$用于Banach Space $ x $的子空间$ y $。我们获得了这些特性的各种新特征。我们讨论了古典Banach空间中的各种示例，上面提到的特性得到满足并失败。据观察，$ C_0 $中的超平面具有属性 - $（Hb）$，并且仅当它是$ m $ -summand时。通过识别$（x \ wideHat {\ otimes}_πy）^*\ cong \ mathcal {l}（x，y^*）$，将$ x，z $作为$ z $的子空间作为$ z $的子空间，我们观察到了$ \ n preseries in $ \ n prionger of und ofers of ymatercal propers ymater { $（x \ wideHat {\ otimes}_πz）$如果$ y $具有属性 - $（su）$ in $ z $。据观察，有限的尺寸子空间$ y $的$ c_0 $具有属性 - $ c_0 $中的$（k-u）$，如果$ y $是理想的，则$ y^*$是$ k $ - stictlictly coNvex cosvex subspace $ \ ell_1 $ \ ell_1 $ \ ell_1 $ \ ell_1 $ g ell_1 $。
In this study, we analyze the various strengthening and weakening of the uniqueness of the Hahn--Banach extension. In addition, we consider the case in which $Y$ is an ideal of $X$. In this context, we study the property-$(U)/ (SU)/ (HB)$ and property-$(k-U)$ for a subspace $Y$ of a Banach space $X$. We obtain various new characterizations of these properties. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is observed that a hyperplane in $c_0$ has property-$(HB)$ if and only if it is an $M$-summand. Considering $X, Z$ as Banach spaces and $Y$ as a subspace of $Z$, by identifying $(X\widehat{\otimes}_πY)^*\cong \mathcal{L}(X,Y^*)$, we observe that an isometry in $\mathcal{L}(X,Y^*)$ has a unique norm-preserving extension over $(X\widehat{\otimes}_πZ)$ if $Y$ has property-$(SU)$ in $Z$. It is observed that a finite dimensional subspace $Y$ of $c_0$ has property-$(k-U)$ in $c_0$, and if $Y$ is an ideal, then $Y^*$ is a $k$-strictly convex subspace of $\ell_1$ for some natural $k$.
